Studies on Asymptotic Properties of Solutions of Differential Equations

本研究的目的是检验1。Lienard系统的轨迹的渐近性态。微分-差分方程系统的稳定域。对于解的振动和其他问题来说，找到决定(1)Lienard系统的轨迹是否接近原点和(2)轨迹是否与垂直等斜线相交的条件是非常重要的。本研究的主要研究者和合作者对上述问题进行了讨论，得到了一些隐式必要和充分条件、显式必要条件和显式充分条件（nonlin . ff. eq . appl .）。得到了Lienard平面上分离矩阵的性质，并给出了具有同斜轨道的Lienard系统周期解的存在性结果，Lienard系统是分离矩阵的典型例子（disc . conte . dyname . syst .）。我们还通过分离矩阵的性质和数目给出了Lienard系统的全局相位画像的完整分类（j.m ath. al. appll .）。此外，我们讨论了二阶Euler型非线性微分方程（Proc.Amer.Math.Soc.）和周期强迫Lienard系统（Proc.Inter.Con.Dynam.Syst.）的振动问题。混乱,Diff.Integ.Eq)。作为对生物模型的应用，我们考虑了一类具有相当一般Holling型功能响应的捕食-食饵系统，给出了该系统恰好有一个稳定极限环的充分必要条件。给出了一类n维微分-差分系统零解指数渐近稳定的充分必要条件。将拟多项式的庞特里亚金判据应用于微分-差分系统的特征方程，证明了我们的结果。在n=1的情况下，我们的结果给出了一个众所周知的标量微分-差分方程的充分必要条件。

The purpose of this research is to examine 1. asymptotic behavior of trajectories of the Lienard system and 2. stability regions for systems of differential-difference equations.1. It is very important for oscillation of solutions and other subjects to find conditions for deciding (1) whether trajectories of the Lienard system approach the origin or not and (2) whether trajectories intersect the vertical isocline. The head investigator and cooperators of this research discussed the above problems and obtained some implicit necessary and sufficient conditions, explicit necessary conditions and explicit sufficient conditions (Nonlin.Diff.Eq.Appl.). As the result, we clarified the properties of separatrices in the Lienard plane and gave existence results on periodic solutions of the lienard system with a homoclinic orbit which is a typical example of separatrix (Disc.Cont.Dynam.Syst.). We also gave a complete classification of global phase portraits of the Lienard system by the properties and the number of separatrices (J.Math.Anal.Appl.). Moreover, we discussed oscillation problems of a second order nonlinear differential equation of Euler type (Proc.Amer.Math.Soc.) and a periodically forced Lienard system (Proc.Inter.Con.Dynam.Syst.Chaos, Diff.Integ.Eq.). As an application to biological model, we considered a predator-prey system with a fairly general functional response of Holling type and presented a necessary and sufficient condition under which this system has exactly one stable limit cycle.2. We gave some necessary and sufficient conditions for the zero solution of an n-dimensional differential-difference system to be exponential asymptotically stable (Funk.Ekvac.). The proof of our results is carried out by an application of Pontryagin criterion for quasi-polynomials to the characteristic equation of the differential-difference system. In case n=1, our results yeild a well-known necessary and sufficient condition for a scalar differential-difference equation.